# Mathematical Physics

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## Mathematical physics

An area of science concerned with the application of mathematical concepts to the physical sciences and the development of mathematical ideas in response to the needs of physics. Historically, the concept of mathematical physics was synonymous with that of theoretical physics. In present-day terminology, however, a distinction is made between the two. Whereas most of theoretical physics uses a large amount of mathematics as a tool and as a language, mathematical physics places greater emphasis on mathematical rigor, and devotes attention to the development of areas of mathematics that are, or show promise to be, useful to physics. The results obtained by pure mathematicians, with no thought to applications, are almost always found to be both useful and effective in formulating physical theories.

Mathematical physics forms the bridge between physics as the description of nature and its structure on the one hand, and mathematics as a construction of pure logical thought on the other. This bridge between the two disciplines benefits and strengthens both fields enormously. *See* Physics, Theoretical physics

The methods employed in mathematical physics range over most of mathematics, the areas of analysis and algebra being the most commonly used. Partial differential equations and differential geometry, with heavy use of vector and tensor methods, are of particular importance in the formulation of field theories, and functional analysis as well as operator theory in quantum mechanics. Group theory has become an especially valuable tool in the construction of quantum field theories and in elementary-particle physics. There has also been an increase in the use of general geometrical approaches and of topology. For solution methods and the calculation of quantities that are amenable to experimental tests, of particular prominence are Fourier analysis, complex analysis, variational methods, the theory of integral equations, and perturbation theory. *See* Variational methods (physics), Vector methods (physics)

## Mathematical Physics

the theory of mathematical models of physical phenomena, which occupies a special place in both mathematics and physics.

Mathematical physics is closely connected with physics inasmuch as it deals with the construction of mathematical models; at the same time it is a branch of mathematics inasmuch as the methods used to investigate the models are mathematical. The concept of mathematical physics also includes those mathematical methods that are used to set up and study mathematical models that describe large classes of physical phenomena.

The methods of mathematical physics as the theories of mathematical models in physics were first developed intensively in I. Newton’s works dealing with the formulation of the foundations of classical mechanics, universal gravitation, and the theory of light. The further development of the methods of mathematical physics and their successful application to a wide range of physical phenomena are associated with J. Lagrange, L. Euler, P. Laplace, J. Fourier, K. Gauss, B. Riemann, and M. V. Ostrogradskii, among others. A. M. Liapunov and V. A. Steklov made major contributions to the development of the methods of mathematical physics.

Beginning in the second half of the 19th century, the methods of mathematical physics were used successfully in studying mathematical models of physical phenomena related to various physical fields and wave functions in electrodynamics, acoustics, the theory of elasticity, hydrodynamics, aerodynamics, and other fields related to the study of physical phenomena in continuous media. The mathematical models of this class of phenomena are described most often by means of partial differential equations called the equations of mathematical physics. In addition, integral and integrodifferential equations, variational and probability theory methods, potential theory, the theory of functions of a complex variable, and many other branches of mathematics are also used in describing the mathematical models of physics. With the rapid development of computer mathematics direct numerical methods based on the use of computers and especially finite-difference methods of solving boundary value problems have acquired particular importance in the investigation of mathematical models in physics. Theoretical investigations in quantum electrodynamics, the axiomatic theory of fields, and many other branches of modern physics have led to the creation of a new class (including the theory of generalized functions and the theory of continuous-spectra operators) of mathematical models constituting an important branch of mathematical physics.

The formulation of the problems of mathematical physics consists in the setting up of mathematical models describing the basic regularities shown by the class of physical phenomena being studied. Such a formulation involves the derivation of equations (differential, integral, integrodifferential, or algebraic) that are satisfied by the quantities characterizing the particular physical process. In so doing we proceed from the fundamental physical laws that take account of only the most significant features of the phenomenon, disregarding its secondary characteristics. These laws are usually the conservation laws (the laws of the conservation of momentum, energy, and the number of particles). Thus, the same mathematical model can be used to describe different physical processes having common features. For example, mathematical questions associated with the simplest hyperbolic equation

which was initially obtained by J. D’Alembert in 1747 to describe the free vibrations of a uniform string, are also pertinent in describing a broad range of wave processes in acoustics, hydrodynamics, and electrodynamics. Similarly, the equation

the boundary value problems for which were originally studied by Laplace (at the end of the 18th century) in connection with the development of the theory of gravitation, subsequently found application in the solution of many problems of electrostatics, the theories of elasticity, and the steady motion of an ideal fluid. An entire class of physical processes corresponds to every mathematical model used in physics.

Another feature of mathematical physics is that many general methods used to solve the problems of mathematical physics were developed from particular methods of solving specific physical problems and in their original form did not have a rigorous mathematical basis or lacked completeness. This is true of such well-known methods as the Ritz and Galerkin methods, the theory of perturbations, Fourier transformations, and the separation of variables. The effective application of all these methods to the solution of specific problems is one reason for their rigorous mathematical substantiation and generalization, which has often led to the emergence of new mathematical disciplines.

The impact of mathematical physics on various branches of mathematics is also evident from the fact that the development of mathematical physics, which reflects the requirements of the natural sciences and technology, entails reorientation of research in some branches of mathematics that have already taken shape. The formulation of the problems of mathematical physics, which was connected with the elaboration of mathematical models of real physical phenomena, has led to a change in the basic approach to the theory of partial differential equations. The theory of boundary value problems emerged, which subsequently made it possible to relate partial differential equations to integral equations and variational methods.

The study of the mathematical models of physics by mathematical methods not only makes it possible to obtain the quantitative characteristics of physical phenomena and to compute with a given degree of accuracy the course of real processes, but also provides the possibility of gaining insight into the very nature of physical phenomena, revealing hidden laws and predicting new effects. The striving for a more detailed study of physical phenomena leads to the formulation of increasingly complex mathematical models. This in turn makes it impossible to use analytic methods to investigate the models. This impossibility arises since the mathematical models of real physical processes generally are nonlinear, that is, are described by nonlinear equations of mathematical physics.

However, direct numerical methods involving the use of computers are used successfully in the detailed investigation of these models. The use of numerical models for typical problems of mathematical physics reduces to replacing the equations of mathematical physics for functions of a continuous argument with algebraic equations for finite-difference functions that are defined in a discrete set of points (in a network). In other words, a discrete analog is introduced in place of a continuous model of the medium. In many cases, the use of numerical methods makes it possible to replace a complex, laborious, and costly physical experiment with a less costly mathematical (numerical) experiment. A mathematical numerical experiment, if it is complete enough, can serve as the basis in the selection of optimal conditions for a real physical experiment, the selection of the parameters of complex physical installations and the determination of the conditions under which new physical effects would appear. Numerical methods thus greatly expand the range of effective use of mathematical models of physical phenomena.

A mathematical model of a physical phenomenon, like any model, cannot convey all the characteristics of the phenomenon. The adequacy of the adopted model for the phenomenon being studied can be established only through practice and a comparison of the results of theoretical investigations of the model adopted with experimental data.

In many cases, the adequacy of the model adopted can be assessed on the basis of the solution of inverse problems of mathematical physics, when the properties of natural phenomena that are inaccessible to direct observation are ascertained from indirect physical manifestations.

A characteristic feature of mathematical physics is the setting up of mathematical models that not only provide a description and explanation of physical principles already established for the range of phenomena under study but also make it possible to predict as yet undiscovered principles. A classic example of such a model is Newton’s theory of universal gravitation, which made it possible not only to explain the motion of bodies in the solar system that were known at the time the theory was developed but also to predict the existence of new planets. On the other hand, new experimental data that appear cannot always be explained within the framework of the model adopted and therefore the development of a more complex model is required.

### REFERENCES

Tikhonov, A. N., and A. A. Samarskii.*Uravneniia matematicheskoi fiziki*, 4th ed. Moscow, 1972.

Vladimirov, V. S.

*Uravneniia matematicheskoi fiziki*, 2nd ed. Moscow, 1971.

Sobolev, S. A.

*Uravneniia matematicheskoi fiziki*. Moscow, 1966.

Courant, R.

*Uravneniia s chastnymi proizvodnymi*. Moscow, 1964. (Translated from English.)

Morse, P. M., and H. Feshbach.

*Melody teoreticheskoi fiziki*, vols. 1-2. Moscow, 1958. (Translated from English.)

A. N. TIKHONOV, A. A. SAMARSKII, and A. G. SVESHNIKOV